Geometric properties of Orlicz spaces equipped with \(p\)-Amemiya norms -- results and open questions (Q2822643)

Given a \(\sigma\)-finite, nonatomic and complete measure \(\mu\) and an Orlicz function \(\Phi\), the author considers the Orlicz space \(L_{\phi}\) of all (equivalence classes of) measurable functions \(x\) for which there is a \(\lambda>0\) with \(I_{\Phi}(\lambda x):=\int\Phi(\lambda x(t))\text{d}\mu(t)<\infty\).NEWLINENEWLINE Given \(p\in[1,\infty]\), this space is equipped with the so called \(p\)-Amemiya norm NEWLINE\[NEWLINE\|x\|_{\Phi,p}:=\inf_{k>0}\frac{1}{k}s_p(I_{\Phi}(kx)),NEWLINE\]NEWLINE where \(s_p(u):=(1+u^p)^{1/p}\) for \(1\leq p<\infty\) and \(s_{\infty}(u):=\max\{1,u\}\).NEWLINENEWLINE All these norms are equivalent and \(\|\cdot\|_{\Phi,\infty}\) coincides with the usual Luxemburg norm, while \(\|\cdot\|_{\Phi,1}\) is the Orlicz norm.NEWLINENEWLINE The space \(L_{\phi}\) equipped with the norm \(\|\cdot\|_{\Phi,p}\) is denoted by \(L_{\Phi,p}\).NEWLINENEWLINE The author reviews several results on Orlicz spaces with \(p\)-Amemiya norms that can be found in the literature. These results concern for example the question for which \(k\) the infimum in the definition of \(\|\cdot\|_{\Phi,p}\) is attained, as well as questions on extreme points/strong extreme points, rotundity/midpoint local uniform rotundity, best approximations and non-squareness in \(L_{\Phi,p}\). Monotonicity properties (in the sense of Banach lattices) as well as the Kadets-Klee property (with respect to convergence in measure) in \(L_{\phi,p}\) are also discussed.NEWLINENEWLINE In the last section of the paper, the author defines a generalisation of \(p\)-Amemiya norms (by replacing the function \(s_p\) with a more general convex function \(s\)) and studies some of their properties.